Information recoverability of noisy quantum states
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Classical information, also known as shadow information, carried by quantum states could be corrupted as they undergo undesirable noises modelled by quantum channels. Due to quantum channels' destructive nature, a fundamental problem in quantum information theory is how well we can retrieve information from noisy quantum states. In this work, we introduce a systematic framework to quantitatively study this problem. First, we prove that classical information can be retrieved if and only if the querying observable is in the image of the whole observable space under the channel's adjoint. Second, we further define the dimension of the evolved observable space as the quantum channel's effective shadow dimension that quantifies the information recoverability of the channel. This quantity can be computed as the rank of a matrix derived from the channel's Kraus operators, and it induces the shadow destructivity as a measure with meaningful properties. Third, we resolve the minimum cost of retrieving a piece of shadow information from the required number of copies of a noisy state. We in particular show that the optimal cost and the corresponding protocol are efficiently computable by semidefinite programming. As applications, we derive the ultimate limits on shadow retrieving costs for generalized amplitude damping channels and mixed Pauli channels and employ the corresponding protocol to mitigate errors in tasks of ground state energy estimation.
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